Simulations of estimating causal effects using undersmoothed HAL
Seraphina Shi
2023-03-17
Simulation 2
Data structure: \(O = (W, A, Z, Y)\)
- U - exogenous variables
- W - baseline covariate that is a measure of body condition
- A - treatment level based on W, continuous between 0 and 5
- Z - intermediate curve based on W and A
- Y - outcome, indicator of an event ?
Underlying data generating process, \(P_{U,X}\)
- Exogenous variables:
- \(U_A \sim Normal(\mu=0, \sigma^2 = 1^2)\)
- \(U_A \sim Normal(\mu=0, \sigma^2 = 2^2)\)
- \(U_Z \sim Uniform(min = 0, max = 1)\)
- \(U_Y \sim Uniform(min = 0, max = 1)\)
- Structural equations F and endogenous variables:
- \(W = U_W\)
- \(A = bound(2 - 0.5W + U_A, min=0, max=5)\)
- \(Z = \mathbf{I}[U_Z < expit(2-W-A)]\)
- \(Y = \mathbf{I}[U_Y < expit(-15 - 4W + 5A + Z(10 + 5W + 2AW))]\)
## W A Z Y
## Min. :-3.73508 Min. :0.0000 Min. :0.0000 Min. :0.0000
## 1st Qu.:-0.64589 1st Qu.:0.6136 1st Qu.:0.0000 1st Qu.:0.0000
## Median : 0.02644 Median :2.0483 Median :0.0000 Median :0.0000
## Mean : 0.01805 Mean :2.1412 Mean :0.4871 Mean :0.4435
## 3rd Qu.: 0.69708 3rd Qu.:3.4273 3rd Qu.:1.0000 3rd Qu.:1.0000
## Max. : 3.88609 Max. :5.0000 Max. :1.0000 Max. :1.0000## Summary of A given W < -1:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 1.493 2.896 2.773 4.225 5.000## Summary of A given -1 < W <= 0:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 0.826 2.253 2.275 3.572 5.000## Summary of A given 0 < W <= 1:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.4018 1.8412 1.9875 3.1971 5.0000## Summary of A given 1 < W:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 0.000 1.367 1.598 2.682 5.000
n = 200
CV HAL
results
## CV selected lambda (from one sample): 0.00342499231220454
1000 repetition
## The average of CV selected lambdas (from 1000 sample): 0.00387846621405548## z=1:
## z=0:
Globally Undersmoothed HAL
results
## Undersmoothed lambda: 0.000799982807508477
## which is 0.233572146909012 * lambda_CV
1000 repetition
## The average of unsersmoothed lambda (from 1000 sample): 0.000851942858017283
## which is 0.217778357408167 * the average of 1000 lambda_CV## z=1:
## z=0:
CV vs Undersmoothing
## z=1:
## z=0:
Oevr a grid of lambda scalers
n = 500
CV HAL
results
## CV selected lambda (from one sample): 0.0016481890921567
1000 repetition
## The average of CV selected lambdas (from 1000 sample): 0.00184274700569968## z=1:
## z=0:
Globally Undersmoothed HAL
results
## Undersmoothed lambda: 0.000267629110962862
## which is 0.162377673918872 * lambda_CV
1000 repetition
## The average of unsersmoothed lambda (from 1000 sample): 0.000393879557187377
## which is 0.213589118948609 * the average of 1000 lambda_CV## z=1:
## z=0:
CV vs Undersmoothing
## z=1:
## z=0:
Oevr a grid of lambda scalers
